Group #90449 — subgroups

Subgroups — C₂×D₇×SD₁₆ #90449

Order 448.

298 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1143	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#1142	C₂	1	1	yes	maximal, characteristic, complemented
6	D₇×SD₁₆	C₂	1	8	yes	maximal, direct product factor
7	C₂×#1132	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₇	—	2	1	no	—
10	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₇	—	2	1	no	—
13	Q₈×D₇	—	2	2	no	—
14	Q₈×D₇	C₂²	1	4	yes	complemented
15	D₄×D₇	—	2	2	no	—
16	D₄×D₇	C₂²	1	4	yes	—
17	C₂×D₂₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₇	—	2	1	no	—
19	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₄	—	2	1	no	—
21	C₂×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
22	C₇×SD₁₆	C₂²	1	4	yes	complemented
23	C₂×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
24	#1143	C₂²	1	4	yes	complemented
25	#1142	C₂²	1	4	yes	complemented
26	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
27	#1132	C₂²	1	4	yes	complemented
28	C₈×D₇	C₂²	1	4	yes	complemented
29	C₂²×SD₁₆	—	7	1	no	maximal, Sylow
30	C₂³×C₇	—	2	1	no	—
31	C₂²×D₇	—	2	1	no	—
32	C₂²×D₇	—	2	4	no	—
33	C₂²×D₇	—	4	4	no	—
34	C₂²×D₇	D₄	1	1	yes	characteristic
35	C₇×Q₈	—	2	1	no	—
36	C₇×Q₈	C₂³	1	2	yes	complemented
37	C₇×D₄	—	2	1	no	—
38	C₇×D₄	C₂³	1	2	yes	—
39	C₂×C₄×C₇	—	2	1	no	—
40	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²⋊D₇	—	2	4	no	—
42	C₂×Dic₇	—	2	1	no	—
43	C₂×Dic₇	D₄	1	1	yes	characteristic
44	D₂₈	—	2	1	no	—
45	D₂₈	C₂³	1	2	yes	—
46	C₄×D₇	—	2	4	no	—
47	C₄×D₇	D₄	1	2	yes	—
48	C₄×D₇	C₂³	1	2	yes	—
49	Dic₁₄	—	2	1	no	—
50	Dic₁₄	—	2	4	no	—
51	Dic₁₄	C₂³	1	2	yes	complemented
52	C₇×C₈	C₂³	1	2	yes	complemented
53	C₇⋊C₈	C₂³	1	2	yes	complemented
54	C₂²×Q₈	—	7	1	no	—
55	C₂²×D₄	—	7	1	no	—
56	C₂×SD₁₆	—	7	1	no	—
57	C₂×SD₁₆	—	7	1	no	—
58	C₂×SD₁₆	—	7	8	no	—
59	C₂×SD₁₆	—	7	1	no	—
60	C₂×SD₁₆	D₇	1	1	yes	characteristic, direct product factor
61	C₂²×C₈	—	7	1	no	—
62	C₂²×C₇	—	2	2	no	—
63	C₂²×C₇	—	4	2	no	—
64	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂×D₇	—	2	2	no	—
66	C₂×D₇	—	4	2	no	—
67	C₂×D₇	—	4	8	no	—
68	C₂×D₇	SD₁₆	1	4	yes	direct product factor
69	C₂×D₇	C₂×D₄	1	2	yes	—
70	C₄×C₇	—	2	2	no	—
71	C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₄×C₇	C₂⁴	1	1	yes	characteristic
73	Dic₇	—	2	2	no	—
74	Dic₇	C₂×D₄	1	2	yes	—
75	C₂⁴	—	14	1	no	—
76	C₂×Q₈	—	7	4	no	—
77	C₂×Q₈	—	7	1	no	—
78	C₂×Q₈	—	14	1	no	—
79	C₂×Q₈	—	14	2	no	—
80	C₂×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
81	C₂×D₄	—	7	4	no	—
82	C₂×D₄	—	7	1	no	—
83	C₂×D₄	—	14	2	no	—
84	C₂×D₄	—	14	1	no	—
85	C₂×D₄	C₂×D₇	1	1	yes	characteristic
86	C₂²×C₄	—	7	1	no	—
87	C₂²×C₄	—	14	1	no	—
88	SD₁₆	—	7	4	no	—
89	SD₁₆	—	7	4	no	—
90	SD₁₆	—	7	4	no	—
91	SD₁₆	C₂×D₇	1	4	yes	direct product factor
92	C₂×C₈	—	7	1	no	—
93	C₂×C₈	—	7	4	no	—
94	C₂×C₈	C₂×D₇	1	1	yes	characteristic, complemented
95	C₂×C₇	—	4	2	no	—
96	C₂×C₇	C₂×SD₁₆	1	2	yes	complemented
97	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
98	D₇	—	4	2	no	—
99	D₇	C₂×SD₁₆	1	4	yes	direct product factor
100	C₂³	—	2	1	no	—
101	C₂³	—	7	1	no	—
102	C₂³	—	14	4	no	—
103	C₂³	—	14	1	no	—
104	C₂³	—	28	4	no	—
105	Q₈	—	2	1	no	—
106	Q₈	—	7	2	no	—
107	Q₈	—	14	4	no	—
108	Q₈	—	14	1	no	—
109	Q₈	C₂²×D₇	1	2	yes	complemented
110	D₄	—	2	1	no	—
111	D₄	—	7	2	no	—
112	D₄	—	14	4	no	—
113	D₄	—	14	1	no	—
114	D₄	C₂²×D₇	1	2	yes	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	7	1	no	—
117	C₂×C₄	—	7	2	no	—
118	C₂×C₄	—	7	2	no	—
119	C₂×C₄	—	14	4	no	—
120	C₂×C₄	—	14	1	no	—
121	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
122	C₈	—	7	2	no	—
123	C₈	C₂²×D₇	1	2	yes	complemented
124	C₇	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂²	—	2	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	7	4	no	—
128	C₂²	—	7	2	no	—
129	C₂²	—	14	2	no	—
130	C₂²	—	28	8	no	—
131	C₂²	—	28	2	no	—
132	C₂²	D₄×D₇	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	7	2	no	—
135	C₄	—	14	2	no	—
136	C₄	D₄×D₇	1	1	yes	characteristic
137	C₄	C₂³×D₇	1	1	yes	characteristic
138	C₂	—	4	2	no	—
139	C₂	—	7	4	no	—
140	C₂	—	28	2	no	—
141	C₂	D₇×SD₁₆	1	2	yes	direct product factor
142	C₂	C₂×D₄×D₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₇

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₇

C₂

yes

maximal, characteristic

C₂×C₇×SD₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#1143

C₂

yes

maximal, characteristic, complemented

C₂×#1142

C₂

yes

maximal, characteristic, complemented

D₇×SD₁₆

C₂

yes

maximal, direct product factor

C₂×#1132

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₇

—

C₂×C₇×Q₈

C₂²

yes

characteristic, complemented

C₂×C₇×D₄

C₂²

yes

characteristic

C₂×C₂²⋊D₇

—

Q₈×D₇

—

Q₈×D₇

C₂²

yes

complemented

D₄×D₇

—

D₄×D₇

C₂²

yes

—

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

—

C₂×C₄×D₇

C₂²

yes

characteristic

C₂×Dic₁₄

—

C₂×Dic₁₄

C₂²

yes

characteristic, complemented

C₇×SD₁₆

C₂²

yes

complemented

C₂×C₇×C₈

C₂²

yes

characteristic, complemented

#1143

C₂²

yes

complemented

#1142

C₂²

yes

complemented

C₂×C₇⋊C₈

C₂²

yes

characteristic, complemented

#1132

C₂²

yes

complemented

C₈×D₇

C₂²

yes

complemented

C₂²×SD₁₆

—

maximal, Sylow

C₂³×C₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

D₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

C₂³

yes

complemented

C₇×D₄

—

C₇×D₄

C₂³

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂²⋊D₇

—

C₂×Dic₇

—

C₂×Dic₇

D₄

yes

characteristic

D₂₈

—

D₂₈

C₂³

yes

—

C₄×D₇

—

C₄×D₇

D₄

yes

—

C₄×D₇

C₂³

yes

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

C₂³

yes

complemented

C₇×C₈

C₂³

yes

complemented

C₇⋊C₈

C₂³

yes

complemented

C₂²×Q₈

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

D₇

yes

characteristic, direct product factor

C₂²×C₈

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×D₄

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

SD₁₆

yes

direct product factor

C₂×D₇

C₂×D₄

yes

—

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

characteristic

C₄×C₇

C₂⁴

yes

characteristic

Dic₇

—

Dic₇

C₂×D₄

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

C₂×D₇

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₇

yes

characteristic, complemented

C₂×C₇

—

C₂×C₇

C₂×SD₁₆

yes

complemented

C₂×C₇

C₂²×D₄

yes

characteristic

D₇

—

D₇

C₂×SD₁₆

yes

direct product factor

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

Q₈

C₂²×D₇

yes

complemented

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

C₂²×D₇

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×D₇

yes

characteristic

122

C₈

—

123

C₈

C₂²×D₇

yes

complemented

124

C₇

C₂²×SD₁₆

yes

characteristic, complemented, Sylow

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

D₄×D₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₄×D₇

yes

characteristic

137

C₄

C₂³×D₇

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

D₇×SD₁₆

yes

direct product factor

142

C₂

C₂×D₄×D₇

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₂×C₇×SD₁₆	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₄×D₇	C₂
hypercenter	C₂×SD₁₆	D₇
hypocenter	C₇	C₂²×SD₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₇

commutator, cosocle

C₄×C₇

C₂⁴

Frattini

C₄

C₂³×D₇

Fitting

C₂×C₇×SD₁₆

C₂

socle

C₂²×C₇

C₂×D₄

prime radical

C₂×D₄×D₇

C₂

hypercenter

C₂×SD₁₆

D₇

hypocenter

C₇

C₂²×SD₁₆

LCS

G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×Q₈×D₇	C₂	semidirect
3. C₂×C₇×SD₁₆	C₂	semidirect
4. C₂×#1143	C₂	semidirect
5. C₂×#1142	C₂	semidirect
6. D₇×SD₁₆	C₂	direct (N × H)
7. C₂×#1132	C₂	semidirect
8. C₂×C₈×D₇	C₂	semidirect
10. C₂×C₇×Q₈	C₂²	semidirect
14. Q₈×D₇	C₂²	semidirect
21. C₂×Dic₁₄	C₂²	semidirect
22. C₇×SD₁₆	C₂²	semidirect
23. C₂×C₇×C₈	C₂²	semidirect
24. #1143	C₂²	semidirect
25. #1142	C₂²	semidirect
26. C₂×C₇⋊C₈	C₂²	semidirect
27. #1132	C₂²	semidirect
28. C₈×D₇	C₂²	semidirect
36. C₇×Q₈	C₂³	semidirect
51. Dic₁₄	C₂³	semidirect
52. C₇×C₈	C₂³	semidirect
53. C₇⋊C₈	C₂³	semidirect
60. C₂×SD₁₆	D₇	direct (N × H)
68. C₂×D₇	SD₁₆	direct (N × H)
80. C₂×Q₈	C₂×D₇	semidirect
91. SD₁₆	C₂×D₇	direct (N × H)
94. C₂×C₈	C₂×D₇	semidirect
96. C₂×C₇	C₂×SD₁₆	semidirect
99. D₇	C₂×SD₁₆	direct (N × H)
109. Q₈	C₂²×D₇	semidirect
123. C₈	C₂²×D₇	semidirect
124. C₇	C₂²×SD₁₆	semidirect
141. C₂	D₇×SD₁₆	direct (N × H)

H ≅ G/N

type

1. C₂×Q₈×D₇

C₂

semidirect

3. C₂×C₇×SD₁₆

C₂

semidirect

4. C₂×#1143

C₂

semidirect

5. C₂×#1142

C₂

semidirect

6. D₇×SD₁₆

C₂

direct (N × H)

7. C₂×#1132

C₂

semidirect

8. C₂×C₈×D₇

C₂

semidirect

10. C₂×C₇×Q₈

C₂²

semidirect

14. Q₈×D₇

C₂²

semidirect

21. C₂×Dic₁₄

C₂²

semidirect

22. C₇×SD₁₆

C₂²

semidirect

23. C₂×C₇×C₈

C₂²

semidirect

24. #1143

C₂²

semidirect

25. #1142

C₂²

semidirect

26. C₂×C₇⋊C₈

C₂²

semidirect

27. #1132

C₂²

semidirect

28. C₈×D₇

C₂²

semidirect

36. C₇×Q₈

C₂³

semidirect

51. Dic₁₄

C₂³

semidirect

52. C₇×C₈

C₂³

semidirect

53. C₇⋊C₈

C₂³

semidirect

60. C₂×SD₁₆

D₇

direct (N × H)

68. C₂×D₇

SD₁₆

direct (N × H)

80. C₂×Q₈

C₂×D₇

semidirect

91. SD₁₆

C₂×D₇

direct (N × H)

94. C₂×C₈

C₂×D₇

semidirect

96. C₂×C₇

C₂×SD₁₆

semidirect

99. D₇

C₂×SD₁₆

direct (N × H)

109. Q₈

C₂²×D₇

semidirect

123. C₈

C₂²×D₇

semidirect

124. C₇

C₂²×SD₁₆

semidirect

141. C₂

D₇×SD₁₆

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Q₈×D₇	2
C₂×D₄×D₇	2
C₂×C₇×SD₁₆	2
C₂×#1143	2
C₂×#1142	2
D₇×SD₁₆	2
C₂×#1132	2
C₂×C₈×D₇	2
C₂²×SD₁₆	7

subgroup

index

C₂×Q₈×D₇

C₂×D₄×D₇

C₂×C₇×SD₁₆

C₂×#1143

C₂×#1142

D₇×SD₁₆

C₂×#1132

C₂×C₈×D₇

C₂²×SD₁₆

Subgroups — C₂×D₇×SD₁₆ #90449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D7×SD16 #90449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₇×SD₁₆ #90449